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- W2808450762 abstract "Gy'arf'as and Lehel and independently Faudree and Schelp proved that in any 2-coloring of the edges of $K_{n,n}$ there exists a monochromatic path on at least $2lceil n/2rceil$ vertices, and this is tight. We prove a stability version of this result which holds even if the host graph is not complete; that is, if $G$ is a balanced bipartite graph on $2n$ vertices with minimum degree at least $(3/4+o(1))n$, then in every 2-coloring of the edges of $G$, either there exists a monochromatic cycle on at least $(1+o(1))n$ vertices, or the coloring of $G$ is close to an extremal coloring -- in which case $G$ has a monochromatic path on at least $2lceil n/2rceil$ vertices and a monochromatic cycle on at least $2lfloor n/2rfloor$ vertices. Furthermore, we determine an asymptotically tight bound on the length of a longest monochromatic cycle in a 2-colored balanced bipartite graph on $2n$ vertices with minimum degree $delta n$ for all $0leq deltaleq 1$." @default.
- W2808450762 created "2018-06-21" @default.
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- W2808450762 date "2018-06-13" @default.
- W2808450762 modified "2023-09-27" @default.
- W2808450762 title "Long monochromatic paths and cycles in 2-colored bipartite graphs" @default.
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